1 Introduction

Let \(Z=L_p([0,{{c}}];{V})\) be a function space, where \(p>1\) and V is a Banach space. Let \({\mathfrak {C}}_t=C([-a,t];{V})\) denotes the set of all V-valued continuous functions defined on \([-a,t]\) with the norm \(\Vert z\Vert _{{\mathfrak {C}}_t}=\sup _{-{a}\le \varrho \le t}\Vert z(\varrho )\Vert \). Consider the semilinear system:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{z}(t)={A}z(t)+\displaystyle \sum _{i=0}^{m}{B}_i{u}(t-a_i)\\ &{}&{}\quad \quad \quad +{F}(t,z_{\nu (t)},{u}(t),{u}(t-{\widehat{a}}_1),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}})), \quad t\in (0,{{c}}];\\ &{}&{}{\dot{z}}(0)=\mu _1;\\ &{}&{}\psi (z)={h},\quad {u}(t)=0, \quad t\in [-a,0];\end{array} \right\} \end{aligned}$$
(1.1)

where the state \(z(t)\in V\) and the control \({u}(\cdot )\in U=L_p([0,{{c}}];V^\prime )\), \(V^\prime \) is another Banach space; \(a_i\) and \({\widehat{a}}_j\), \(j=1,2,\ldots ,{\widehat{m}}\) are fixed delays, such that \(0=a_0<a_1<a_2<\cdots<a_m<c\), \(0<{\widehat{a}}_1<{\widehat{a}}_2<\cdots<{\widehat{a}}_{{\widehat{m}}}<{{c}}\) and \(a=\max \{a_m,{\widehat{a}}_{{\widehat{m}}}\}\). A generates a strongly continuous cosine family \(\{{\mathcal {C}}(t):t\in {\mathbb {R}}\}\) on V; \({B}_0,B_1,\ldots ,B_m\) are continuous linear maps from \({V}^\prime \) to V; \(\nu :[0,{{c}}]\rightarrow [0,{{c}}]\) is non-expansive and nondecreasing map satisfying \(\nu (t)\le t\); the function \(z_{\nu (t)}\in {\mathfrak {C}}_0\) is defined by \(z_{\nu (t)}(\varsigma )=z(\nu (t)+\varsigma ),\quad \varsigma \in [-a,0]\); and \({F}:[0,{{c}}]\times {\mathfrak {C}}_0\times \underbrace{{V}^\prime \times {V}^\prime \times \cdots \times {V}^\prime }_{({\widehat{m}}+1)~\text{ times }}\rightarrow {V}\) is nonlinear. \(\psi \) and h together represent the nonlocal delay condition.

Controllability is a fundamental property of dynamical systems, which was introduced by Kalman [1] in 1960. He discussed the controllability of deterministic linear systems. Roughly speaking, a dynamical system is called controllable during some bounded time interval, over a space V, if we can steer that system from any initial state to any final state in V during that time interval, using a set of control functions. There are several types of controllability, namely; total state controllability, complete state controllability, approximate controllability, trajectory controllability, null controllability, interior controllability, exact controllability, constrained controllability, partial approximate controllability, etc. Some results on various types of controllability can be seen in [2,3,4,5,6,7,8,9,10]. According to mathematical viewpoint, it is very important to recognize the difference between approximate and exact controllability. Approximate controllability empowers to steer the system to any given neighborhood of any final state, but by exact controllability, we mean that the system can be steered to any given final state. Obviously, the condition of exact controllability is necessarily stronger than approximate controllability. Approximate controllability makes free to steer a system to states belonging to a dense subset of the state space. Therefore, it is reasonable to investigate the approximate controllability of a system.

There are many natural incidents which involve a noteworthy memory effect. Such real-life problems can be modeled by retarded differential equations. For example, problems of economics, physical sciences, chemical sciences, biosciences, and medicine are affected by their history results. Therefore, it is very important to discuss the controllability of a retarded system. For any real world phenomena, it is always better to represent an abstract model with delays appearing in more than one step. For example, in patient’s treatment, there may be delays in the availability of doctors and in the diagnosis. In the abstract system (1.1), the delay term \(z_{\nu (t)}\in {\mathfrak {C}}_0\) represents the notion of generalized delay, where delays occur in two steps. For some results on the controllability of retarded systems of fractional and integer order, we refer [11,12,13,14] and some references therein.

Existence of solutions for nonlocal abstract Cauchy problems has been proven by Byszewski and Lakshmikantham [15]. Results for the existence of solutions of nonlocal semilinear systems with and without impulses can be seen in [16,17,18]. The controllability results for nonlocal semilinear systems of fractional and integer order have been proved in many articles (see [13, 14, 19,20,21,22] and references therein). Controllability of linear systems with control delays was proved by Klamka [23]. Constrained controllability results for semilinear systems with multiple delays in control can be seen in [24,25,26]. Klamka [27] proved the stochastic controllability of system with multiple delays in control. Shen [28] proved the stochastic controllability results for impulsive systems with multiple delays in control. However, to the best of our knowledge, there is no result on approximate controllability of nonlocal semilinear retarded systems of second order with multiple delays in control and locally Lipschitz non-linearity.

The article is structured like this: Sect. 2 contains the preliminaries. Existence and controllability results are derived in Sects. 3 and 4, respectively. An example is provided to illustrate the theory in Sect. 5.

2 Preliminaries

We present some definitions and preliminary facts which are to be used in forthcoming sections. First, we define sine and cosine family, because mild solutions for the systems of second order are defined in terms of these families (for details, see [29, 30]). For this, let \({\mathcal {B}}({V})\) denotes the set of bounded linear maps from V to itself.

Definition 2.1

A family of operators \(\left\{ {\mathcal {C}}(t):{t\in {\mathbb {R}}}\right\} \subset {\mathcal {B}}({V})\) is called strongly continuous cosine family if:

  1. (i)

    \({\mathcal {C}}(0)z=z\) for all \(z\in V\);

  2. (ii)

    \(2{\mathcal {C}}(s){\mathcal {C}}(t)={\mathcal {C}}(s-t)+{\mathcal {C}}(s+t)\) for all \(s,t\in {\mathbb {R}}\);

  3. (iii)

    \({\mathcal {C}}(t)\) is strongly continuous in t.

The sine family \(\left\{ {\mathcal {S}}(t):{t\in {\mathbb {R}}}\right\} \) associated with \(\left\{ {\mathcal {C}}(t):{t\in {\mathbb {R}}}\right\} \) is defined as:

$$\begin{aligned} {\mathcal {S}}(t)z=\int _0^t{\mathcal {C}}(s)z\mathrm{d}s,\quad z\in {V},\quad t\in {\mathbb {R}}. \end{aligned}$$

Throughout this article, we suppose that \(\Vert {\mathcal {C}}(t)\Vert \le \eta _1\) and \(\Vert {\mathcal {S}}(t)\Vert \le \eta _2\), \(0\le t\le c\), where \(\eta _1\) and \(\eta _2\) are constants. We also utilize the set:

$$\begin{aligned} {V}_1=\left\{ z\in {V}: {\mathcal {C}}(t)z\quad \text{ is } \text{ continuously } \text{ differentiable }\right\} . \end{aligned}$$

Lemma 2.2

[29] If A generates a strongly continuous cosine family \(\left\{ {\mathcal {C}}(t):{t\in {\mathbb {R}}}\right\} \), then:

  1. (i)

    \({\mathcal {S}}(t)=-{\mathcal {S}}(-t)\) and \({\mathcal {C}}(t)={\mathcal {C}}(-t)\) for all \(t\in {\mathbb {R}}\);

  2. (ii)

    \({\mathcal {S}}(s),{\mathcal {S}}(t),{\mathcal {C}}(s)\) and \({\mathcal {C}}(t)\) commute for all \(s,t\in {\mathbb {R}}\);

  3. (iii)

    \(2{\mathcal {S}}(s){\mathcal {C}}(t)={\mathcal {S}}(s-t)+{\mathcal {S}}(s+t)\) for all \(s,t\in {\mathbb {R}}\);

  4. (iv)

    \({\mathcal {C}}(s){\mathcal {S}}(t)+{\mathcal {C}}(t){\mathcal {S}}(s)={\mathcal {S}}(s+t)\) for all \(s,t\in {\mathbb {R}}\);

  5. (v)

    \(2A{\mathcal {S}}(s){\mathcal {S}}(t)={\mathcal {C}}(s+t)-{\mathcal {C}}(s-t)\) for all \(s,t\in {\mathbb {R}}\);

  6. (vi)

    if \(z\in V_1\), then \({\mathcal {S}}(t)z\in D(A)\) and \(\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {C}}(t)z=A{\mathcal {S}}(t)z\);

  7. (vii)

    if \(z\in V_1\), then \({\mathcal {S}}(t)z\in D(A)\) and \(\frac{\mathrm{d}^2}{\mathrm{d}t^2}{\mathcal {S}}(t)z=A{\mathcal {S}}(t)z\).

Definition 2.3

Suppose \(\wp \in {\mathfrak {C}}_0\) satisfies \(\psi (\wp )={h}\). A function \(z(\cdot )\in {\mathfrak {C}}_{{c}}\) is called the mild solution of (1.1) if:

$$\begin{aligned} z(t)={\left\{ \begin{array}{ll} {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\displaystyle \int _0^t{\mathcal {S}}(t-s)\left( \displaystyle \sum _{i=0}^m{B}_i{u}(s-a_i)\right) \,\mathrm{d}s\\ \quad \quad \quad +\displaystyle \int _0^t{\mathcal {S}}(t-s){F}(s,z_{\nu (s)},{u}(s),{u}(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\,\mathrm{d}s, &{}\quad t\in (0,{{c}}];\\ \wp (t), &{}\quad t\in [-a,0].\end{array}\right. } \nonumber \\ \end{aligned}$$
(2.1)

Moreover, if \(\wp (0)\in {V}_1\), then \({\dot{z}}(t)\) is continuous on [0, c], and it is given by:

$$\begin{aligned} {\dot{z}}(t)&=A{\mathcal {S}}(t)\wp (0)+{\mathcal {C}}(t)\mu _1+\int _0^t{\mathcal {C}}(t-s) \left( \displaystyle \sum _{i=0}^m{B}_i{u}(s-a_i)\right) \,\mathrm{d}s\\&\quad +\int _0^t{\mathcal {C}}(t-s){F}(s,z_{\nu (s)},{u}(s),{u}(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\,\mathrm{d}s. \end{aligned}$$

The systems

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{z}(t)={A}z(t)+\displaystyle \sum _{i=0}^{m}{B}_i{u}(t-a_i), \quad t\in (0,{{c}}];\\ &{}&{}{\dot{z}}(0)=\mu _1,\quad z(0)=\wp (0);\\ &{}&{}{u}(t)=0,\quad t\in [-a,0]\end{array} \right\} \end{aligned}$$
(2.2)

and

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{z}(t)={A}z(t)+{B}_0{u}(t), \quad t\in (0,{{c}}];\\ &{}&{}{\dot{z}}(0)=\mu _1,\quad z(0)=\wp (0)\end{array} \right\} \end{aligned}$$
(2.3)

are corresponding linear systems with delays and without delay, respectively, if \(\psi (\wp )=h\).

Definition 2.4

The system given by (1.1) is said to be approximately controllable on [0, c], if, for every given \(\epsilon >0\) and a final state \(z_c\in {V}\), one can find a control \(u\in U\), such that the mild solution z(t) corresponding to the control u satisfies \(\Vert z(c)-z_c\Vert \le \epsilon \).

3 Existence of Mild Solution

To discuss the existence result, we suppose the following:

\(H_1\):

\(\psi :C([-a,0];{V}_1)\rightarrow C([-a,0];{V}_1)\) and there exists a unique function \(\wp \in C([-a,0];{V}_1)\) satisfying \(\psi (\wp )={h}\).

\(H_2\):

F is continuous in t and locally Lipschitz in z that is there exists a constant \(\lambda _r>0\) satisfying:

$$\begin{aligned} \left\| F(t,z_1,u_0,{u_1},\ldots ,u_{{\widehat{m}}})-F(t,{z_2},u_0,{u_1},\ldots ,u_{{\widehat{m}}})\right\| \le \lambda _r\left\| z_1-{z_2}\right\| _{{\mathfrak {C}}_0} \end{aligned}$$

for all \(t\in [0,{{c}}]\); \(z_\ell \in {\mathfrak {C}}_0\) with \(\Vert z_\ell \Vert _{{\mathfrak {C}}_0}\le r\), \(\ell =1,2\) and \({u_j}\in {V}^\prime \), \(j=0,1,2,\ldots {\widehat{m}}\).

\(H_3\):

There exists a \(\kappa >0\) satisfying:

$$\begin{aligned} \Vert F(t,z,u_0,u_1,\ldots ,u_{{\widehat{m}}})\Vert \le \kappa (1+\Vert z\Vert _{{\mathfrak {C}}_0}+\Vert u_0\Vert +\Vert u_1\Vert +\cdots +\Vert u_{{\widehat{m}}}\Vert ) \end{aligned}$$

for all \(t\in [0,{{c}}]\); \(z\in {\mathfrak {C}}_0\), and \(u_j\in {V}^\prime \), \(j=0,1,2,\ldots ,{\widehat{m}}\).

First, we deduce the following lemma:

Lemma 3.1

Let z(t) be continuous on \([-a,c)\). If \(k_1\) and \(k_2\) be two positive constants, such that:

$$\begin{aligned} \Vert z(t)\Vert \le k_1+k_2\int _0^t\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_0}\,\mathrm{d}s\quad \forall t\in [0,c). \end{aligned}$$

Then,

$$\begin{aligned} \Vert z(t)\Vert \le (M_\wp +k_1)e^{k_2c}\quad \forall t\in [0,c), \end{aligned}$$

where \(z(t)=\wp (t)\) for \(t\in [-a,0]\) and \(M_\wp =\sup _{t\in [-a,0]}\Vert \wp (t)\Vert \).

Proof

Let \({{\widehat{t}}}\in [0,c)\) be arbitrary. Then,

one can find a \(t^*\in [-a,{{\widehat{t}}}]\) satisfying:

$$\begin{aligned} \sup _{\varsigma \in [-a,0]}\Vert z(\nu ({{\widehat{t}}})+\varsigma )\Vert =\Vert z(t^*)\Vert . \end{aligned}$$

Now if \(t^*\in [-a,0]\), then:

$$\begin{aligned} \sup _{\varsigma \in [-a,0]}\Vert z(\nu ({{\widehat{t}}})+\varsigma )\Vert&=\Vert z(t^*)\Vert \\&\le M_\wp \\&< M_\wp +k_1+k_2\int _0^{{{\widehat{t}}}}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_{0}}\,\mathrm{d}s. \end{aligned}$$

If \(t^*\in (0,{{\widehat{t}}}]\), then:

$$\begin{aligned} \sup _{\varsigma \in [-a,0]}\Vert z(\nu ({{\widehat{t}}})+\varsigma )\Vert&=\Vert z(t^*)\Vert \\&\le k_1+k_2\int _0^{t^*}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_{0}}\,\mathrm{d}s \\&\le M_\wp +k_1+k_2\int _0^{{{\widehat{t}}}}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_{0}}\,\mathrm{d}s. \end{aligned}$$

Thus:

$$\begin{aligned} \Vert z({{\widehat{t}}})\Vert \le \sup _{\varsigma \in [-a,0]}\Vert z(\nu ({{\widehat{t}}})+\varsigma )\Vert \le M_\wp +k_1+k_2\int _0^{{{\widehat{t}}}}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_{0}}\,\mathrm{d}s \\ \Rightarrow \Vert z(t)\Vert \le M_\wp +k_1+k_2\int _0^{t}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_{0}}\,\mathrm{d}s\quad \forall \, t\in [0,c). \end{aligned}$$

In view of Gronwall’s inequality, we obtain:

$$\begin{aligned} \Vert z(t)\Vert \le (M_\wp +k_1)e^{k_2c}\quad \forall \, t\in [0,c). \end{aligned}$$

\(\square \)

Theorem 3.2

Under hypotheses \([H_1]\)–\([H_3]\), the system (1.1) has a unique mild solution for each \({u}(\cdot )\in U\) and \(\mu _1\in {V}\).

Proof

Let \(0<{{c}}_1<{{c}}\) and \(\max \big \{\Vert {B}_0\Vert ,\Vert {B}_1\Vert ,\ldots ,\Vert B_m\Vert \big \}\le M_{B}\). Define a mapping \({\Phi }:{\mathfrak {C}}_{{{c}}_1}\rightarrow {\mathfrak {C}}_{{{c}}_1}\) by:

$$\begin{aligned} ({\Phi } z)(t)= {\left\{ \begin{array}{ll} {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\displaystyle \int _0^t{\mathcal {S}}(t-s)\left( \displaystyle \sum _{i=0}^m{B}_i{u}(s-a_i)\right) \,\mathrm{d}s\\ \quad +\displaystyle \int _0^t{\mathcal {S}}(t-s){F}(s,z_{\nu (s)},{u}(s),{u}(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\,\mathrm{d}s, &{}\quad t\in (0,{{c}}_1];\\ \wp (t),&{}\quad t\in [-a,0];\end{array}\right. } \nonumber \\ \end{aligned}$$
(3.1)

and consider the ball:

$$\begin{aligned} B_{r_0}=\left\{ z(\cdot )\in {\mathfrak {C}}_{{{c}}_1}:\Vert z\Vert _{{\mathfrak {C}}_{{{c}}_1}}\le r_0, z(0)=\wp (0)~\text{ and }~{\dot{z}}(0)=\mu _1\right\} . \end{aligned}$$

Then, for any \(z(\cdot )\in B_{r_0}\) and \(0\le s\le {{c}}_1\):

$$\begin{aligned} \Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_0}=\sup _{\varsigma \in [-a,0]}\Vert z(\nu (s)+\varsigma )\Vert \le \sup _{\varrho \in [-a,{{c}}_1]}\Vert z(\varrho )\Vert \le r_0. \end{aligned}$$

Thus:

$$\begin{aligned} \Vert ({\Phi } z)(t)\Vert&\le \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +\eta _2 M_B \left( \int _{0}^{t} \displaystyle \sum _{i=0}^m\Vert u(s-a_i)\Vert \,\mathrm{d}s\right) \\&\quad +\eta _2\int _{0}^{t}\Vert F(t,z_{\nu (s)},u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\\&\quad -F(t,0,u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\Vert \,\mathrm{d}s\\&\quad +\eta _2\int _{0}^{t}\Vert F(t,0,u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\Vert \,\mathrm{d}s.\end{aligned}$$

Using \([H_2]\) and \([H_3]\), we obtain:

$$\begin{aligned} \Vert ({\Phi } z)(t)\Vert&\le \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U +\eta _2\lambda _{r_0}\int _{0}^{t}\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_0}\,\mathrm{d}s\\&\quad +\eta _2\kappa \int _0^t\left[ 1+\Vert u(s)\Vert +\Vert u(s-{\widehat{a}}_1)\Vert +\cdots +\Vert u(s-{\widehat{a}}_{{\widehat{m}}})\Vert \right] \mathrm{d}s\\&\le \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U\\&\quad +\eta _2\lambda _{r_0}r_0{{c}}_1+\eta _2\kappa \left[ {{c}}_1+({\widehat{m}}+1){{{c}}_1}^{1-\frac{1}{p}}\Vert u\Vert _U\right] \\&= \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U\\&\quad +\eta _2\left[ \lambda _{r_0}r_0{{c}}_1+\kappa \left( {{c}}_1+({\widehat{m}}+1){{{c}}_1}^{1-\frac{1}{p}}\Vert u\Vert _U\right) \right] . \end{aligned}$$

Now, choosing \(r_0=2\left[ \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U\right] +1\) and \({{c}}_1\) small enough, such that:

$$\begin{aligned}&\eta _2\left[ \lambda _{r_0}r_0{{c}}_1+\kappa \left( {{c}}_1+({\widehat{m}}+1){{{c}}_1}^{1-\frac{1}{p}}\Vert u\Vert _U\right) \right] \le \eta _1\Vert \wp (0)\Vert \\&\quad +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U+1. \end{aligned}$$

Then:

$$\begin{aligned} \Vert ({\Phi } z)(t)\Vert&\le 2\left[ \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B{{c}}^{1-\frac{1}{p}}\Vert u\Vert _U\right] +1\\&=r_0. \end{aligned}$$

Therefore, \({\Phi }\) maps \(B_{r_0}\) into itself.

Now, take \(z_1,z_2\in B_{r_0}\), then:

$$\begin{aligned} \Vert ({\Phi } z_1)(t)-({\Phi } z_2)(t)\Vert&\le \eta _2\int _{0}^{t}\Vert F(s,(z_1)_{\nu (s)},u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\\ {}&\quad -F(s,(z_2)_{\nu (s)},u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}}))\Vert \,\mathrm{d}s\\&\le \eta _2\lambda _{r_0}\int _0^t\Vert (z_1)_{\nu (s)}-(z_2)_{\nu (s)}\Vert _{{\mathfrak {C}}_0}\,\mathrm{d}s\\&\le \eta _2\lambda _{r_0}t\Vert z_1-z_2\Vert _{{\mathfrak {C}}_{{c}_1}}. \end{aligned}$$

Repeating the above process, one can obtain:

$$\begin{aligned} \Vert ({\Phi }^{n} z_1)(t)-({\Phi }^{n} z_2)(t)\Vert&\le \frac{\left( \eta _2\lambda _{r_0}{{t}}\right) ^{n}}{{n}!}\Vert z_1-z_2\Vert _{{\mathfrak {C}}_{{c}_1}}\\&\le \frac{\left( \eta _2\lambda _{r_0}{{c}}\right) ^{n}}{{n}!}\Vert z_1-z_2\Vert _{{\mathfrak {C}}_{{c}_1}}, \end{aligned}$$

which shows that \({\Phi }^{n}\) is a contraction map for sufficiently large value of n. By Banach fixed point theorem, \({\Phi }\) has a fixed point in \(B_{r_0}\). Hence, (2.1) is a mild solution on \([-a,{{c}}_1]\). In similar way, the existence of mild solution on \([{{c}}_1,{{c}}_2]\), where \({{c}}_1<{{c}}_2\), can be shown. Applying the above technique, one can deduce that (2.1) is a mild solution on the maximal existing interval \([-a,{{c}}^*)\), \({{c}}^*\le {{c}}\). Next, we show the boundedness of solution. Clearly, z(t) is bounded on \([-a,0]\). Now, for \(t\in [0,{{c}}^*)\):

$$\begin{aligned} \Vert z(t)\Vert&\le \eta _1\Vert \wp (0)\Vert +\eta _2\Vert \mu _1\Vert +(m+1)\eta _2M_B\int _0^t\Vert u(s)\Vert \,\mathrm{d}s\\ {}&\quad +\eta _2\kappa \int _0^t\big (1+\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_0}+({\widehat{m}}+1)\Vert u(s)\Vert \big )\,\mathrm{d}s\\&\le \eta _1\Vert \wp (0)\Vert \\&\quad +\eta _2\left[ \Vert \mu _1\Vert +\big ((m+1)M_B+({\widehat{m}}+1)\kappa \big ){{c}}^{1-\frac{1}{p}}\Vert u\Vert _U+\kappa {{c}}\right] \\&\quad +\eta _2\kappa \int _0^t\Vert z_{\nu (s)}\Vert _{{\mathfrak {C}}_0}\,\mathrm{d}s. \end{aligned}$$

By Lemma 3.1, we have:

$$\begin{aligned} \Vert z(t)\Vert\le & {} \Big [\eta _1\Vert \wp (0)\Vert +\eta _2(\Vert \mu _1\Vert +\big ((m+1)M_B\\&+({\widehat{m}}+1)\kappa \big ){{c}}^{1-\frac{1}{p}}\Vert u\Vert _U+\kappa {{c}}\big )+M_\wp \Big ]e^{\eta _2\kappa {{c}}}, \end{aligned}$$

which shows that z(t) is bounded on \([-a,{{c}}^*)\), and hence, it is defined on \([-a,{{c}}]\).

For uniqueness, suppose \(z_1\) and \(z_2\) be two solutions of (1.1) for the same control function u. Then, \(z_1(t)=z_2(t)=\wp (t)\) for \(t\in [-a,0]\). Now, for \(t\in [0,{{c}}]\), let:

$$\begin{aligned} a^*=\max \big \{\Vert z_1\Vert _{{\mathfrak {C}}_{{c}}},\Vert z_2\Vert _{{\mathfrak {C}}_{{c}}}\big \}. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z_1(t)-z_2(t)\Vert _{V}&\le \eta _2\int _0^t\big \Vert F\big (s,(z_1)_{\nu (s)},u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}})\big )\\ {}&\quad -F\big (s,(z_2)_{\nu (s)},u(s),u(s-{\widehat{a}}_1),\ldots ,u(s-{\widehat{a}}_{{\widehat{m}}})\big )\big \Vert \,\mathrm{d}s\\&\le \eta _2\lambda _{a^*}\int _0^t\big \Vert (z_1)_{\nu (s)}-(z_2)_{\nu (s)}\big \Vert _{{\mathfrak {C}}_0} \mathrm{d}s\\&\le \eta _2\lambda _{a^*}\int _0^{{c}}\big \Vert (z_1)_{\nu (s)}-(z_2)_{\nu (s)}\big \Vert _{{\mathfrak {C}}_0} \mathrm{d}s. \end{aligned}$$

Therefore:

$$\begin{aligned} \Vert (z_1)_{\nu (t)}-(z_2)_{\nu (t)}\Vert _{{\mathfrak {C}}_0}\le \eta _2\lambda _{a^*}\int _0^{{c}}\big \Vert (z_1)_{\nu (s)}-(z_2)_{\nu (s)}\big \Vert _{{\mathfrak {C}}_0} \mathrm{d}s. \end{aligned}$$

By Gronwall’s inequality, we obtain \((z_1)_{\nu (t)}=(z_2)_{\nu (t)}\) for all \(t\in [0,{{c}}]\), and consequently, \(z_1=z_2\). This completes the proof. \(\square \)

Remark 3.3

It is notable that if the map \(\psi \) is not injective, then the system (1.1) may have more than one solution for a fixed control \(u(\cdot )\in U\).

4 Approximate Controllability

The subsequent discussion needs the following hypotheses:

\(H_4\):

The system (2.3) is approximately controllable.

\(H_5\):

There exists a function \(q(\cdot ) \in L_1[0, c ]\) satisfying:

$$\begin{aligned} \Vert F(t,z,u_0,u_1,\ldots ,u_{{\widehat{m}}})\Vert \le q(t) \end{aligned}$$

for all \((t,z,u_0,u_1,\ldots ,u_{{\widehat{m}}})\in [0, c ]\times {\mathfrak {C}}_0\times {V}^\prime \times {V}^\prime \times \cdots \times {V}^\prime \). First, we prove the controllability of the corresponding linear system using the technique similar to [32].

Theorem 4.1

Under hypotheses \([H_1]\) and \([H_4]\), the corresponding linear delay system (2.2) is approximately controllable.

Proof

Set \({{c}}=a_{m+1}\) and \(r=\min \{a_1,a_2-a_1,a_3-a_2,\ldots ,a_{m+1}-a_m\}\). Since \(0=a_0<a_1<a_2<\cdots<a_m<a_{m+1}\). Therefore, for each \(a_{i+1}\), one can find a positive integer \(n_i\) and a constant \(\alpha _i\in [0,r)\) satisfying \(a_{i+1}=a_i+n_ir+\alpha _i\), \(i=1,2,\ldots ,m\).

Case 1: If \(\alpha _1,\alpha _2,\ldots ,\alpha _m\) are positive.

Let \({\tilde{z}}_0; {\tilde{z}}_{11},{\tilde{z}}_{12}\ldots ,{\tilde{z}}_{1n_1},{\tilde{z}}_{1\overline{n_1+1}};{\tilde{z}}_{21},{\tilde{z}}_{22},\ldots ,{\tilde{z}}_{2n_2},{\tilde{z}}_{2\overline{n_2+1}};\cdots ;{\tilde{z}}_{m1},{\tilde{z}}_{m2},\ldots ,{\tilde{z}}_{mn_m},z_{c}\) be given in V , where \({z}_{{{c}}}\) is the final state. Consider the system:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{\xi }(t)={A} \xi (t)+{B}_0{u}(t), \quad t\in (0,a_1];\\ &{}&{}{\dot{\xi }}(0)={\xi }_1=\mu _1;\\ &{}&{}\xi (0)=\wp (0).\end{array} \right\} \end{aligned}$$
(4.1)

Let \(\epsilon >0\) be given. Set \({\tilde{\xi }}_0={\tilde{z}}_0\). By \([H_4]\), one can find a control \({u}_0\), such that the mild solution \( \xi (t)\) of (4.1) given by:

$$\begin{aligned} \xi (t)={\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _0^t {\mathcal {S}}(t-s){B}_0{u}_0(s)\,\mathrm{d}s,\quad 0<t\le a_1, \end{aligned}$$

satisfies \(\Vert \xi (a_1)-{\tilde{\xi }}_0\Vert \le \epsilon \).

Let

$$\begin{aligned} w_0(t)= {\left\{ \begin{array}{ll} 0,&{}\quad t\in [-a,0];\\ {u}_0(t),&{}\quad t\in [0,a_1] \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} z(t)= & {} {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s)\left( \sum _{i=0}^m{B}_iw_0(s-a_i)\right) \,\mathrm{d}s,\quad \\&0<t\le a_1. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z(a_1)-{\tilde{z}}_0\Vert&=\Vert \xi (a_1)-{\tilde{\xi }}_0\Vert \\&\le \epsilon . \end{aligned}$$

Denote \( \xi (a_1)\) by \(\xi _{a_1}\) and \({\dot{\xi }}(a_1)\) by \({\dot{\xi }}_{a_1}\) and consider the system:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{\xi }(t)={A} \xi (t)+{B}_0{u}(t), \quad t\in (a_1,a_1+r];\\ &{}&{}{\dot{\xi }}(a_1)={\dot{\xi }}_{a_1};\\ &{}&{}\xi (a_1)=\xi _{a_1}.\end{array} \right\} \end{aligned}$$
(4.2)

Set \({\tilde{\xi }}_{11}={\tilde{z}}_{11}-\chi _{a_1+r}\), where \(\chi _{a_1+r}=\int _0^{a_1+r}{\mathcal {S}}(a_1+r-s)\big (\sum _{i=1}^m{B}_iw_0(s-a_i)\big )\,\mathrm{d}s=\int _0^{a_1+r}{\mathcal {S}}(a_1+r-s){B}_1w_0(s-a_1)\,\mathrm{d}s\) is known.

Again by \([H_4]\), one can find a control \({u}_{11}\), such that the mild solution \( \xi (t)\) of (4.2) given by:

$$\begin{aligned} \xi (t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s){B}_0{u}_{11}(s)\,\mathrm{d}s,\quad a_1<t\le a_1+r, \end{aligned}$$

satisfies \(\Vert \xi (a_1+r)-{\tilde{\xi }}_{11}\Vert \le \epsilon \).

Let

$$\begin{aligned} w_{11}(t)= {\left\{ \begin{array}{ll} w_0(t),&{}\quad t\in [0,a_1];\\ {u}_{11}(t),&{}\quad t\in (a_1,a_1+r] \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} z(t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s)\left( \sum _{i=0}^m{B}_iw_{11}(s-a_i)\right) \!\mathrm{d}s,\quad \! a_1<t\le a_1+r. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z(a_1+r)-{\tilde{z}}_{11}\Vert&=\Vert \xi (a_1+r)+\chi _{a_1+r}-{\tilde{z}}_{11}\Vert \\ {}&=\Vert \xi (a_1+r)-{\tilde{\xi }}_{11}\Vert \\&\le \epsilon . \end{aligned}$$

Continuing in similar fashion, at the \((n_1+2)\)th step, we get:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{\xi }(t)={A} \xi (t)+{B}_0u(t), \quad t\in (a_1+n_1r,a_2];\\ &{}&{}{\dot{\xi }}(a_1+n_1r)={\dot{\xi }}_{a_1+n_1r};\\ &{}&{}\xi (a_1+n_1r)=\xi _{a_1+n_1r}.\end{array} \right\} \end{aligned}$$
(4.3)

Set \({\tilde{\xi }}_{1\overline{n_1+1}}={\tilde{z}}_{1\overline{n_1+1}}-\chi _{a_2}\), where \(\chi _{a_2}=\int _0^{a_2}{\mathcal {S}}(a_2-s)\big (\sum _{i=1}^mB_iw_{1n_1}(s-a_i)\big )\,\mathrm{d}s=\int _0^{a_2}{\mathcal {S}}(a_2-s)B_1w_{1n_1}(s-a_1)\,\mathrm{d}s\) is known. Then, one can find a control \({u}_{1\overline{n_1+1}}\), such that the mild solution \( \xi (t)\) of (4.3) given by:

$$\begin{aligned} \xi (t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s){B}_0{u}_{1\overline{n_1+1}}(s)\,\mathrm{d}s,\quad a_1+n_1r<t\le a_2, \end{aligned}$$

satisfies \(\Vert \xi (a_2)-{\tilde{\xi }}_{1\overline{n_1+1}}\Vert \le \epsilon \).

Let

$$\begin{aligned} w_{1\overline{n_1+1}}(t)= {\left\{ \begin{array}{ll} w_{1n_1}(t),&{}\quad t\in (a_1+\overline{n_1-1}r,a_1+n_1r];\\ {u}_{1\overline{n_1+1}}(t),&{}\quad t\in (a_1+n_1r,a_2] \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned}&z(t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s)\left( \sum _{i=0}^m{B}_iw_{1\overline{n_1+1}}(s-a_i)\right) \,\mathrm{d}s,\\&\quad a_1+n_1r<t\le a_2. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z(a_2)-{\tilde{z}}_{1\overline{n_1+1}}\Vert&=\Vert \xi (a_2)+\chi _{a_2}-{\tilde{z}}_{1\overline{n_1+1}}\Vert \\ {}&=\Vert \xi (a_2)-{\tilde{\xi }}_{1\overline{n_1+1}}\Vert \\&\le \epsilon . \end{aligned}$$

Repeating the above process, at the last step that is \((n_1+n_2+\cdots +n_m+m+1)\)th step, we get:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\ddot{\xi }(t)={A} \xi (t)+{B}_0u(t), \quad t\in (a_m+n_mr,{{c}}];\\ &{}&{}{\dot{\xi }}(a_m+n_mr)={\dot{\xi }}_{a_m+n_mr};\\ &{}&{} \xi (a_m+n_mr)=\xi _{a_m+n_mr}.\end{array} \right\} . \end{aligned}$$
(4.4)

Set \({\tilde{\xi }}_{{{c}}}={\tilde{z}}_{{{c}}}-\chi _{{{c}}}\), where \(\chi _{{{c}}}=\int _0^{{{c}}}{\mathcal {S}}({{c}}-s)\big (\sum _{i=1}^mB_iw_{mn_m}(s-a_i)\big )\,\mathrm{d}s\) is known from previous step. Then, one can find a control \({u}_{m\overline{n_m+1}}\), such that the mild solution \( \xi (t)\) of (4.4) given by:

$$\begin{aligned} \xi (t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s){B}_0{u}_{m\overline{n_m+1}}(s)\,\mathrm{d}s,\quad a_m+n_mr<t\le {{c}}, \end{aligned}$$

satisfies \(\Vert \xi ({{c}})-{\tilde{\xi }}_{{{c}}}\Vert \le \epsilon \).

Let

$$\begin{aligned} w_{m\overline{n_m+1}}(t)= {\left\{ \begin{array}{ll} w_{mn_m}(t),&{}\quad t\in (a_m+\overline{n_m-1}r,a_m+n_mr];\\ {u}_{m\overline{n_m+1}}(t),&{}\quad t\in (a_m+n_mr,{{c}}] \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned}&z(t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s)\left( \sum _{i=0}^m{B}_iw_{m\overline{n_m+1}}(s-a_i)\right) \,\mathrm{d}s,\\&\quad a_m+n_mr<t\le {{c}}. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z({{c}})-{z}_{{c}}\Vert&=\Vert \xi ({{c}})+\chi _{{{c}}}-{z}_{{{c}}}\Vert \\ {}&=\Vert \xi ({{c}})-{\tilde{\xi }}_{{{c}}}\Vert \\&\le \epsilon . \end{aligned}$$

Now, define the control \(w(\cdot )\) on \([-a,{{c}}]\) as:

$$\begin{aligned} w(t)= {\left\{ \begin{array}{ll} w_0,&{}\quad t\in [-a,a_1];\\ w_i(t),&{} \quad t\in (a_i,a_{i+1}],\quad i=1,2,\ldots ,m; \end{array}\right. } \end{aligned}$$

where:

$$\begin{aligned} w_i(t)= {\left\{ \begin{array}{ll} w_{ij}(t),&{}\quad t\in (a_i+\overline{j-1}r,a_i+jr],\quad j=1,2,\ldots ,n_i;\\ u_{i\overline{n_i+1}}(t),&{}\quad t\in (a_i+n_ir,a_{i+1}]. \end{array}\right. } \end{aligned}$$

Then:

$$\begin{aligned} z(t)= {\mathcal {C}}(t)\wp (0)+{\mathcal {S}}(t)\mu _1+\int _{0}^{t} {\mathcal {S}}(t-s)\left( \sum _{i=0}^m{B}_iw(s-a_i)\right) \,\mathrm{d}s,\quad 0<t\le {{c}} \end{aligned}$$

is the mild solution of (2.2) for the control function w, and it satisfies \(\Vert z({{c}})-z_{{c}}\Vert \le \epsilon \). For other cases, the proof is similar. \(\square \)

Theorem 4.2

Under hypotheses \([H_1]\)–\([H_5]\), the semilinear system (1.1) is approximately controllable.

Proof

Since \(q(\cdot )\in L_1[0,{{c}}]\), one can find an increasing sequence \(\langle {{c}}_n \rangle \) in [0, c], such that \({{c}}_n \rightarrow {{c}}\) and:

$$\begin{aligned} \int _{{{c}}_n}^{{{c}}}q(t)\,\mathrm{d}t\rightarrow 0, \quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

Now, by approximate controllability of (2.2), for any given \(\epsilon >0\) and \(z_{{c}}\in {{V}}\), one can find a control \({\tilde{u}}_0\in U\) satisfying:

$$\begin{aligned} \left\| z_{{c}}-{\mathcal {C}}({{c}})\wp (0)-{\mathcal {S}}({{c}})\mu _1-\int _{0}^{{{c}}}{\mathcal {S}}({{c}}-s)\left( \sum _{i=0}^m{B}_i{\tilde{u}}_0(s-a_i)\right) \,\mathrm{d}s\right\| \le \frac{\epsilon }{2}. \end{aligned}$$

Denote \(z_1=z({{c}}_1,\wp ,{\tilde{u}}_0)\) and \({\dot{z}}_1={\dot{z}}({{c}}_1,\wp ,{\tilde{u}}_0)\), where \(z(t,\wp ,{\tilde{u}}_0)\) is the mild solution of (1.1) for the control \({\tilde{u}}_0\). Again by approximate controllability of (2.2), one can find a control \({\tilde{u}}_1\in L_p([{{c}}_1,{{c}}];{V}')\) satisfying:

$$\begin{aligned} \left\| z_{{c}}-{\mathcal {C}}({{c}}-{{c}}_1)z_1-{\mathcal {S}}({{c}}-{{c}}_1){\dot{z}}_1-\int _{{{c}}_1}^{{{c}}}{\mathcal {S}}({{c}}-s)\left( \sum _{i=0}^m{B}_i{\tilde{u}}_1(s-a_i)\right) \,\mathrm{d}s\right\| \le \frac{\epsilon }{2}. \end{aligned}$$

Define:

$$\begin{aligned} {\tilde{w}}_1(t)= {\left\{ \begin{array}{ll} {\tilde{u}}_0(t),&{}\quad t\in [0,{{c}}_1);\\ {\tilde{u}}_1(t),&{}\quad t\in [{{c}}_1,{{c}}]. \end{array}\right. } \end{aligned}$$

Clearly, \({\tilde{w}}_1(\cdot )\in U\). Continuing in this manner, one can obtain three sequences \(z_n,{\tilde{u}}_n\) and \({\tilde{w}}_n\), such that \({\tilde{u}}_n(\cdot )\in L_p([{{c}}_n,{{c}}];{V}')\), \({\tilde{w}}_n(\cdot )\in U\) given by:

$$\begin{aligned} {\tilde{w}}_n(t)= {\left\{ \begin{array}{ll} {\tilde{u}}_{n-1}(t),&{}\quad t\in [0,{{c}}_n);\\ {\tilde{u}}_n(t),&{}\quad t\in [{{c}}_n,{{c}}] \end{array}\right. } \end{aligned}$$

and \(z_n=z({{c}}_n,\wp ,{\tilde{u}}_{n-1})\), \({\dot{z}}_n={\dot{z}}({{c}}_n,\wp ,{\tilde{u}}_{n-1})\) with:

$$\begin{aligned} \left\| z_{{c}}-{\mathcal {C}}({{c}}-{{c}}_n)z_n-{\mathcal {S}}({{c}}-{{c}}_1){\dot{z}}_n-\int _{{{c}}_n}^{{{c}}}{\mathcal {S}}({{c}}-s)\left( \sum _{i=0}^m{B}_i{\tilde{u}}_n(s-a_i)\right) \,\mathrm{d}s\right\| \le \frac{\epsilon }{2}. \end{aligned}$$

Let \(z(t,\wp ,{\tilde{w}}_n)\) be the mild solution of (1.1) associated with \({\tilde{w}}_n\). Denote:

$$\begin{aligned} G_{F,B}(s)=\sum _{i=0}^mB_i{\tilde{w}}_n(s-a_i)+F(s,z_{\nu (s)},{\tilde{w}}_n(s),{\tilde{w}}_n(s-{\widehat{a}}_1),\ldots ,{\tilde{w}}_n(s-{\widehat{a}}_{{\widehat{m}}})). \end{aligned}$$

Then:

Now:

Hence, the system (1.1) is approximately controllable. \(\square \)

5 Example

Consider the semilinear wave equation for \(0\le x\le 1\) on [0, c]:

$$\begin{aligned} \left. \begin{array}{rcl} &{}&{}\frac{\partial ^2 {\tilde{z}}(t,x)}{\partial t^2}=\frac{\partial ^2{\tilde{z}}(t,x)}{\partial x^2}+\displaystyle \sum _{i=0}^m{u}(t-a_i,x)\\ &{}&{}\quad \quad \quad \quad ~+{F}(t,{\tilde{z}}(\nu (t)+\varsigma ,x),{u}(t,x),{u}(t-{\widehat{a}}_1,x),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}})), \quad t\in [0,{{c}}];\\ &{}&{}\frac{\partial {\tilde{z}}}{\partial x}(t,0)=\frac{\partial {\tilde{z}}}{\partial x}(t,1)=0, \quad t\in [0,{{c}}];\\ &{}&{}\frac{\partial {\tilde{z}}}{\partial t}(0,x)={\tilde{\mu }}_1(x);\\ &{}&{}\displaystyle \sum _{j=1}^n\beta _j{\tilde{z}}(t_j,x)={\tilde{z}}_0(x); \end{array} \right\} \nonumber \\ \end{aligned}$$
(5.1)

where \(-a\le t_1<t_2<\cdots <t_n\le 0\). The above equation can be converted in the abstract form (1.1), if we make the following setting:

(i) Define \(A:D(A)\subseteq {V}\rightarrow {V}\) by \(Ay=\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\), where \({V}=L_2[0,1]\) and:

$$\begin{aligned} D(A)= & {} \left\{ y\in {V}:y,~y_x\quad \text{ are } \text{ absolutely } \text{ continuous },\quad \right. \\&\left. y_{xx}\in {V}\quad \text{ and }\quad y_x(0)=0=y_x(1)\right\} . \end{aligned}$$

For \(\ell =1,2,\ldots \); take \(f_\ell (x)={2^{1/2}}\cos l\pi x\) and \(\sigma _\ell =(\ell \pi )^2\). Clearly, \(0,\sigma _1,\sigma _2,\ldots \) are eigenvalues of A with eigenfunctions \(1,f_1,f_2,\ldots \); respectively. Also, the set \(\{1,f_1,f_2,\ldots \}\) is an orthonormal basis for V. Then:

$$\begin{aligned} Ay=-\sum _{\ell =1}^{\infty }(\ell \pi )^2\left\langle y,f_\ell \right\rangle f_\ell ,\quad y\in {V}, \end{aligned}$$

and A generates the cosine family \(\left\{ {\mathcal {C}}(t):t\in {\mathbb {R}}\right\} \) given by:

$$\begin{aligned} {\mathcal {C}}(t)y=\sum _{\ell =1}^{\infty }\cos (\ell \pi t)\left\langle y,f_\ell \right\rangle f_\ell ,\quad y\in {V}, \end{aligned}$$

with corresponding sine family:

$$\begin{aligned} {\mathcal {S}}(t)y=\sum _{\ell =1}^{\infty }\frac{1}{\ell \pi }\sin (\ell \pi t)\left\langle y,f_\ell \right\rangle f_\ell ,\quad y\in {V}. \end{aligned}$$

Now define:

$$\begin{aligned} {V}'=\left\{ {u}(t)\in L_2[0,1]\,\bigg |\,{u}(t)=\sum _{\ell =2}^{\infty }\delta _\ell f_\ell \quad \text{ with }\quad \sum _{\ell =2}^{\infty }\delta _\ell ^2<\infty \right\} \end{aligned}$$

and \(B_i{u}(t)={u}(t)\).

  1. (ii)

    \(\nu (t)=\frac{t^3}{1+c^3}\), \(t\in [0,{{c}}]\), which satisfies \(\nu (t)\le t\) and \({\tilde{z}}_{\nu (t)}(\varsigma ,x)={\tilde{z}}\left( \frac{t^3}{1+c^3}+\varsigma ,x\right) \).

  2. (iii)

    \(\psi (z)(t)={\mathcal {P}}(z)\) for \(z\in {\mathfrak {C}}_0\), \(t\in [-a,0]\); \({h}(t)=z_0={\tilde{z}}_0(x)\), where \({\mathcal {P}}:{\mathfrak {C}}_0\rightarrow {V}\) is such that \({\mathcal {P}} (z)=\sum _{j=1}^{n}\beta _jz(t_i)\) and \(z(t_j)={\tilde{z}}(t_j,x)\). We take \(\wp (t)=\sum _{j=1}^{n}\wp (t_j)\), where \(\wp (t_j)=\frac{1}{\beta _j}\frac{1}{n}z_0\), and then, for each \(t\in [-a,0]\):

    $$\begin{aligned} \psi (\wp )(t)={\mathcal {P}}(\wp )=\sum _{j=1}^{n}\beta _j\wp (t_j)=\sum _{j=1}^{n}\beta _j\frac{1}{\beta _j}\frac{1}{n}z_0=z_0={h}(t). \end{aligned}$$

    If we take

    $$\begin{aligned}&F(t,z_{\nu (t)},{u}(t),{u}(t-{\widehat{a}}_1),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}}))\\ {}&\quad =\left( \frac{t\left\| z_{\nu (t)}\right\| _{{\mathfrak {C}}_0}}{1+\left\| z_{\nu (t)}\right\| _{{\mathfrak {C}}_0}}f_3(x)+\frac{t^2(\Vert {u}(t)\Vert +\Vert {u}(t-{\widehat{a}}_1)\Vert +\cdots +\Vert u(t-{\widehat{a}}_{{\widehat{m}}})\Vert )}{1+\Vert {u}(t)\Vert +\Vert {u}(t-{\widehat{a}}_1)\Vert +\cdots +\Vert u(t-{\widehat{a}}_{{\widehat{m}}})\Vert }f_4(x)\right) , \end{aligned}$$

    then:

    $$\begin{aligned}&\Vert F(t,z_{\nu (t)},{u}(t),{u}(t-{\widehat{a}}_1),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}}))\Vert \\ {}&\quad \le 2^{1/2}(t+t^2)\\ {}&\quad \le 2^{1/2}(c+c^2)\left( 1+\Vert z_{\nu (t)}\Vert _{{\mathfrak {C}}_0}+\Vert u(t)\Vert +\Vert u(t-{\widehat{a}}_1)\Vert +\cdots +\Vert u(t-{\widehat{a}}_{{\widehat{m}}})||\right) . \end{aligned}$$

    Hence, \([H_3]\) and \([H_5]\) are satisfied.

Also:

$$\begin{aligned}&\Vert F(t,(z_1)_{\nu (t)},{u}(t),{u}(t-{\widehat{a}}_1),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}}))\\&\qquad -F(t,(z_2)_{\nu (t)},{u}(t),{u}(t-{\widehat{a}}_1),\ldots ,u(t-{\widehat{a}}_{{\widehat{m}}}))\Vert \\ {}&\quad \le c\left\| (z_1)_{\nu (t)}-(z_2)_{\nu (t)}\right\| _{{\mathfrak {C}}_0} \end{aligned}$$

for any \((z_1)_{\nu (t)},(z_2)_{\nu (t)}\in {\mathfrak {C}}_0\) and \({u}(t)\in {V}^\prime \).

Hence, \([H_2]\) is satisfied. The linear part of (5.1) is approximately controllable (in fact, it is exactly controllable) [31]. Thus, by previous theorem, the system (5.1) is controllable.

6 Conclusion

In this article, approximate controllability for retarded systems of second order with control delays and nonlocal conditions has been discussed by assuming that the nonlinear term is locally Lipschitz which is a weaker condition than Lipschitz continuity. Using fixed point approach, the existence and uniqueness results have been derived. The system with delay effect appearing in two steps is considered. Here, the controllability results have been derived without assuming the inclusion relation among the range sets of the operators. However, conditions \([H_3]\) and \([H_5]\) are very strong, and may not be easily satisfied in many practical problems. For this reason, an study on approximate controllability of the same system without assuming the conditions \([H_3]\) and \([H_4]\) is a matter of next investigation.